Let
$A={E}_{\mathfrak{1}}\times {E}_{\mathfrak{2}}$ be the product of two
elliptic curves over
$\mathbb{Q}$, each
having a rational
$\mathfrak{5}$torsion
point
${P}_{i}$. Set
$B:=A\u2215\langle \left({P}_{\mathfrak{1}},{P}_{\mathfrak{2}}\right)\rangle $. In this paper
we give an algorithm to decide whether the order of the TateShafarevich group of the abelian
surface
$B$ is square
or five times a square, under the assumptions that we can find a basis for the MordellWeil
groups of
${E}_{\mathfrak{1}}$ and
${E}_{\mathfrak{2}}$ and that the
TateShafarevich groups of
${E}_{\mathfrak{1}}$
and
${E}_{\mathfrak{2}}$
are finite.
We considered all pairs
$\left({E}_{\mathfrak{1}},{E}_{\mathfrak{2}}\right)$
with prescribed bounds on the conductor and the coefficients in
a minimal Weierstrass equation. In total we considered around
$\mathfrak{2}\mathfrak{0}.\mathfrak{0}$ million abelian
surfaces, of which
$\mathfrak{4}\mathfrak{9}.\mathfrak{1}\mathfrak{6}\%$
have TateShafarevich groups of nonsquare order.
